多元复合函数求导法则的条件分析  

Conditional analysis of derivation rule of multivariate composite function

作　　者：王瑞星 吴克坚 WANG Ruixing;WU Kejian(School of Basic Medical Science,Air Force Medical University,Xi'an Shaanxi 710032)

机构地区：[1]空军军医大学基础医学院,陕西西安710032

出　　处：《辽宁师专学报（自然科学版）》2023年第3期6-9,79,共5页Journal of Liaoning Normal College(Natural Science Edition)

基　　金：空军军医大学基础医学院“人才建设行动计划”公共教研室骨干人才(2021-02)。

摘　　要：针对在教学过程中普遍存在的学生对多元复合函数求导法则成立的条件理解不够深入的问题,首先通过举反例的形式说明了一元复合函数求导法则的条件不可以直接应用到多元复合函数求导,然后重点对比分析了多元复合函数求导法则成立的4种不同条件,并给出了条件最强的定理的证明过程.通过对4种不同条件的分析,可以促使学生在学习过程中加深对多元复合函数求导法则的理解,强化对多元函数可导、可微、偏导数连续之间关系的掌握,也为教师有效地开展课堂教学提供参考.As the generalization of the derivation chain rule of unary composite function,the derivation rule of multivariate composite function occupies an important position in differential calculus of multivariate function and even in the whole higher mathematics.In view of the prob-lems that the students generally do not have a deep understanding of the conditions for the deriva-tion rule of multivariate composite function in the teaching process,the paper first explains that the conditions for the derivation of unary composite function cannot be directly applied to the der-ivation of multivariate function composition by taking counter examples,and then focuses on the comparative analysis of four different conditions for the derivation rule of multivariate composite function,and gives the proof process of the theorem with the strongest conditions.Through the analysis on four different conditions,the students can deepen their understanding of the deriva-tion rule of multivariate composite function in the learning process,strengthen their understand-ing of the relationships among the derivability,differentiability and partial derivative continuity of multivariate functions,and also provide reference for the teachers to carry out the classroom teaching effectively.

关 键 词：多元复合函数 链式法则 可微 

分 类 号：O172.1[理学—数学]